Discriminants of Chebyshev radical extensions

Gassert, T. Alden

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 607-633.

Soit $t$ un nombre entier et $\ell \ne 2$ un nombre premier. Soit $\Phi \left(x\right)=T_{\ell }^n\left(x\right)-t$ la composition $n$-fois du polynôme de Tchebychev de degré $\ell$ décalée de $t$. Supposant que ce polynôme est irréductible, soit $K=\mathbb{Q}\left(\theta \right)$, où $\theta$ est une racine de $\Phi$. Nous appliquons un théorème de Dedekind en conjonction avec des résultats antérieurs de l’auteur afin d’obtenir des conditions sur $t$ qui assurent que $K$ soit monogène. Pour d’autres valeurs de $t$, nous appliquons un théorème de Guàrdia, Montes, et Nart pour obtenir une formule pour le discriminant de $K$ et calculons une base intègrale de l’anneau des entiers ${𝒪}_K$.

Let $t$ be any integer and fix an odd prime $\ell$. Let $\Phi \left(x\right)=T_{\ell }^n\left(x\right)-t$ denote the $n$-fold composition of the Chebyshev polynomial of degree $\ell$ shifted by $t$. If this polynomial is irreducible, let $K=\mathbb{Q}\left(\theta \right)$, where $\theta$ is a root of $\Phi$. We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on $t$ that ensure $K$ is monogenic. For other values of $t$, we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of $K$ and compute an integral basis for the ring of integers ${𝒪}_K$.

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